Solve equations and simplify expressions

In algebra 1 we are taught that the two rules for solving
equations are the addition rule and the multiplication/division
rule.
The addition rule for equations tells us that the same quantity
can be added to both sides of an equation without changing the
solution set of the equation.

Example

Adding 12 to each side of the equation on the first line of the
example is the first step in solving the equation. We did not
change the solution by adding 12 to each side since both the second
and third equations have the same solution. Equations that have the
same solution sets are called equivalent equations.

The multiplication/division rule for equations tell us that
every term on both sides of an equation can be multiplied or
divided by the same term (except zero) without changing the
solution set of the equation.

Example

When we simplify an expression we operate in the following
order:

1. Simplify the expressions inside
parentheses, brackets, braces and fractions bars.
2. Evaluate all powers.
3. Do all multiplications and division from left
to right.
4. Do all addition and subtractions from left to
right.

A useful rule is the denominator-numerator rule which states
that the denominator and numerator may be multiplied by the same
quantity without changing the value of the fraction.

Example

First we simplify the expression inside the parentheses by
evaluating the powers and then do the subtraction within it.

We then remove the parentheses and multiply both the denominator
and the numerator by √2.

As a last step we do all multiplications and division from left
to right.